S_klogW said:
Bessel functions are a family of special functions that arise in many areas of mathematics and physics. They are named after the mathematician Friedrich Bessel and are often used to solve differential equations that arise in problems involving circular or cylindrical symmetry.
An integral of Bessel functions is an expression that involves integrating a Bessel function or a combination of Bessel functions with respect to some variable. These integrals often arise in the solution of differential equations, and can be evaluated using various techniques such as contour integration and series expansions.
Bessel functions have many important applications in science, particularly in physics and engineering. They are used in the study of wave phenomena, such as sound and electromagnetic waves, and in the analysis of heat transfer and diffusion processes. They also have applications in signal processing, optics, and quantum mechanics.
Bessel functions are closely related to other special functions, such as the hypergeometric function, the confluent hypergeometric function, and the gamma function. In fact, many of these functions can be expressed in terms of Bessel functions, and vice versa. This allows for the simplification of complicated mathematical expressions and makes Bessel functions a powerful tool in solving problems in various fields.
Some common examples of integrals involving Bessel functions include the Hankel transform, the Fourier-Bessel transform, and the Mellin-Barnes integral. These integrals have applications in the fields of signal processing, image processing, and quantum mechanics, among others. They also have various properties and identities that can be used to simplify or evaluate them in specific cases.